1. Introduction 1
1.6. Techniques
1.6.2. Monitoring Enzyme Kinetics
1.6.2.3. Michealis-Menten Equation
Named in honour of L. Michaelis and M. L. Menten (Michaelis and Menten, 1913), this equation was derived in 1902 by A. Brown (Brown, 1902) and V. Henri (Henri, 1902) and is based in its final version on the steady state assumption of G. E.
Briggs and J. B. S. Haldane (Briggs and Haldane, 1925), derived in 1925. The equation
states that the simplest type of reaction catalyzed by an enzyme (E) with a single substrate (S), assuming irreversible conversion to product (P) is:
1 2
E + S ←⎯⎯kk−1→ ES ⎯⎯→k E + P --- (1.42) To describe this reaction, a differential equation can be formulated for each component:
d [S] / dt = -k1 [S] [E] + k-1[ES] --- (1.43) d [E] / dt = -k1 [S] [E] + (k-1 + k2) [ES] --- (1.44) d [ES] / dt = k1 [S] [E] – (k-1 +k2) [ES] --- (1.45)
d [P] / dt = k2 [ES] = v0 --- (1.46)
It is not possible to derive a simple rate equation from these relationships and therefore simplifications were sought. They concentrate on the plausible assumption that the binding equilibrium set up in the first part of the reaction should be fast compared with the following catalytic step, so that the rate constants would be related according to k1 ~ k-1 > k2. In the original assumption the catalytic constant k2 should be so low that it does not influence the binding equilibrium. However, since the reaction velocity depends directly on the amount of ES (v0 = k2 [ES]), v0 is a direct measure of ES and therefore can be used to determine the dissociation constant according to the law of mass action:
1 1
[S][E]
d [ES]
K k k
= − = --- (1.47) In their more differentiated view, Briggs and Haldane recognized that k2 cannot be completely neglected; rather, formation (due to k1) and decay (due to k-1 and k2) of the enzyme-substrate complex ES compensate for one another, so that [ES] remains constant:
d [ES] / dt = 0. This state, however, is maintained for only a limited time and was designated ‘steady state’ by the authors, to differentiate it from real equilibrium. For this time period the rate equation for the overall reaction can be simplified. The differential equation for the enzyme-substrate complex (1.42) can thus be simplified:
d [ES] / dt = k1[S] [E] – (k-1 + k2) [ES] = 0 --- (1.48) Taking into account that the total amount of enzyme used in the assay consists of free and complex-bound enzyme, [E] 0 = [E] + [ES], this equation can be rearranged:
1 0
1 1 2
[S][E]
[ES] [S] ( )
k
k k− k
= + + --- (1.49) and inserted into the expression for the velocity, dividing the nominator and denominator by k1:
2 0
0 2
1 2 1
[S][E]
[P] [ES]
[S] (( ) / ) k
v d k
dt k− k k
= = =
+ + ---- (1.50) The constant term of the three rate constants in the denominator is replaced by one kinetic constant, the Michaelis constant: Km = (k-1 + k2) / k1. Since during the experiment the total amount of enzyme should remain constant, it is combined with the catalytic rate constant to give the maximum velocity Vmax = k2 [E]0, which is reached when all enzyme molecules present in the assay are taking part in catalysis.
m ax m 0
V [S]
[S] K v =
+ --- (1.51) This is the final form of the Michaelis-Menten equation. Although the catalytic constant k2 (generally designated kcat) is a characteristic constant for a given enzyme, the maximum velocity Vmax depends on both the immediate amount and activity of enzyme present in the reaction mixture and cannot easily be compared between different assay conditions (especially when carried out in different laboratories). For its exact determination, both the molarity and the specific activity of the enzyme must be known.
In contrast to this, the Michaelis constant is independent of enzyme amount and activity, and corresponding values should be obtained under similar test conditions.
In comparison to the dissociation constant Kd, the Michaelis constant Km is extended by the catalytic constant k2. Consequently, it is similar to Kd only if k2 becomes very small compared with k1 and k-1, but it differs considerably from the dissociation constant the closer k2 becomes to k1 and k-1. But since in most cases k2 is rather small, the contribution of the binding constant to the Michaelis constant dominates the contribution of the catalytic constant and in these cases the Michaelis constant can be regarded essentially as an indication of affinity. Like the dissociation constant, Km has the dimension of concentration (M), but the catalytic constant k2 is a first-order rate constant and has the dimension of s-1, and the maximum velocity Vmax is in units of concentration per time (M s-1). To check the validity of Michaelis-Menten equation for a special
enzyme reaction and to determine the Michaelis constant and the maximum velocity, the reaction rate must be determined with different amounts of substrate. The actual enzyme velocity is obtained from the slope of the initial, linear part of the progress curve, as d [P] / dt or –d [S] / dt (M s-1 or µmol min-1). Plotting these values against the initial substrate concentration should yield a hyperbolic saturation curve as shown in Figure 1.12.
Figure 1.12: (a) Saturation curve according to the Michaelis-Menten equation (b) Determination of maximum velocity Vmax and the Michaelis constant Km are shown.
The velocity approaches a plateau at infinite substrate concentration, since for [S] >> Km, the latter can be neglected and the Michaelis-Menten equation reduces to v0 = Vmax. For [S] = Km, the equation becomes v0 = Vmax/2, and the substrate at half-maximum velocity has the value of Km. Because of this relationship the substrate concentrations for studying the Michaelis-Menten kinetics should be chosen within the range of the Michaelis constant, preferentially from one order of magnitude below to one order of magnitude above Km.
If we examine the v0 versus [S] curve, we find three distinct regions where the velocity responds in a characteristic way to increasing [S] (Figure 1.12). At very low substrate concentrations (eg. [S] < 0.01 Km), the v0 versus [S] curve is essentially linear;
that is, the velocity (for all practical purposes) is directly proportional to the substrate concentration. This is the region of first-order kinetics. At very high substrate concentration (eg. [S] > 100 Km), the velocity is essentially independent of the substrate
concentration. This is the region of zero-order kinetics. At intermediate substrate concentrations, the relationship between v0 and [S] follows neither first-order nor zero- order kinetics.